AbstractBackground: Real-time PCR analysis is a sensitive DNA quantification technique that has recently gained considerable attention in biotechnology, microbiology and molecular diagnostics. Although, the cycle-threshold (Ct) method is the present "gold standard", it is far from being a standard assay. Uniform reaction efficiency among samples is the most important assumption of this method. Nevertheless, some authors have reported that it may not be correct and a slight PCR efficiency decrease of about 4% could result in an error of up to 400% using the Ct method. This reaction efficiency decrease may be caused by inhibiting agents used during nucleic acid extraction or copurified from the biological sample. We propose a new method (Cy0) that does not require the assumption of equal reaction efficiency between unknowns and standard curve.
Conclusion: Our results demonstrate that Cy0 represents a significant improvement over the standard methods for obtaining a reliable and precise nucleic acid quantification even in sub-optimal amplification conditions overcoming the underestimation caused by the presence of some PCR inhibitors.
BackgroundIn the last few years, the real-time polymerase chain reaction (PCR) has rapidly become the most widely used technique in modern molecular biology [1-4].
Experimental designThe absolute quantification method relies on the comparison of distinct samples, such as the comparison of a biological sample with a standard curve of known initial concentration .
Quantitative Real-Time PCRThe DNA standard consisted of a pGEM-T (Promega) plasmid containing a 104 bp fragment of the mitochondrial gene NADH dehydrogenase 1 (MT-ND1) as insert. This DNA fragment was produced by the ND1/ND2 primer pair (forward ND1: 5 ' -ACGCCATAAAACTCTTCACCAAAG-3 ' and reverse ND2: 5 ' -TAGTAGAAGAGCGATGGTGAGAGCTA-3 ' ). This plasmid was purified using the Plasmid Midi Kit (Qiagen) according to the manufacturer ' s instructions. The final concentration of the standard plasmid was estimated spectophotometrically by averaging three replicate A260 absorbance determinations.
Description of the SCF methodFluorescence readings were used to fit the following 4-parameter sigmoid function using nonlinear regression analysis:
Description of the Cy0 methodThe Cy0 value is the intersection point between the abscissa axis and tangent of the inflection point of the Richards curve obtained by the non-linear regression of raw data (Fig. 1).
Figure 1Example of modelling PCR amplification with a 5-parameter Richards function Effectiveness of this model is illustrated by the predicted values generated by Eq. 3 (open circles) that agree with the observed fluorescence (dot and line). Curve-fitting of experimentally derived fluorescence dataset to Eq. 3 generates values for the kinetic parameters from which the inflection point (solid black rhombus) and the slope of the curve can be derived. The quantitative entity Cy0 (solid black dot), used in the proposed method, shows the cross point between the x-axis and the tangent crossing the inflection point of real-time PCR fluorescence curve.
Statistical data analysisNonlinear regressions (for 4-parameter sigmoid and 5-parameter Richards functions) were performed determining unweighted least squares estimates of parameters using the Levenberg-Marquardt method. Accuracy was calculated using the following equation: , where was the relative error, while and were the estimated and the true number of DNA molecules for each combination of input DNA (nDna) and amplification mix percentage (%mix) used in the PCR. Precision was calculated as: , where was the coefficient of variation, and were the mean and the standard deviation for each combination of nDna and %mix. In order to verity that the Richards curves, obtained by nonlinear regression of fluorescence data, were not significantly different from the sigmoidal curves, the values of d parameter were compared to the expected value d= 1, using t test for one sample. For each combination of nDna, %mix, the t values were calculated as follow: , where and were the mean and the standard error of d values for each combination of nDna and %mix, with p(t) < 0.05 for significance level. values were reported using 3-d scatterplot graphic, a complete second order polinomial regression function was shown to estimate the trend of accuracy values. where also reported using 3-d contour plots using third-order polynomials spline fitting. All elaborations and graphics were obtained using Excel (Microsoft), Statistica (Statsoft) and Sigmaplot 10 (Systat Software Inc.).
Experimental system 1: reduction of amplification mix percentageWith our experimental set up, the mean PCR reaction efficiency was 88% under optimal amplification conditions and slightly decreased in the presence of smaller amplification mix up to 84%. Moreover, for decreasing amplification mix amounts, the PCR reaction efficiencies showed higher dispersion levels than optimal conditions leading to increasing quantitative errors (Variation Interval, VI100%= 1.921-1.852 and VI60%= 1.903-1.776; Fig. 2). Subsequently, the fluorescence data obtained in these reactions were used to calculate the initial DNA amount using four different procedures: SCF, Ct, Cp and Cy0.
Precision and accuracy of the SCF methodPrevious studies have shown that the SCF approach can lead to quantification without prior knowledge of amplification efficiency [18, 19, 26]; therefore, we evaluated the performance of this method on our data set. To assess the effect of unequal efficiencies on accuracy, the calculated input DNA, expressed as molecular number, was compared to the expected value obtaining the relative error (RE). The precision was further evaluated measuring the variation coefficient (CV%) of the estimated initial DNA in the presence of different PCR efficiencies and input DNA.
Figure 2Estimation of PCR efficiency using LinReg method Efficiency values were determined from 420 independent reactions using a combination of 3.14 x 107 x 3.14 x 101 DNA molecules as starting template and amplification mix quantities ranging from 60% to 100%. The graph shows the distribution of PCR efficiencies in relation to the percentage of amplification mix used in the reaction. The solid black squares (?) represent the mean of each distribution.
The Cy0 methodThe SCF model assumes that the fluorescence signal is proportional to the amount of product, which is often the case for SYBR-Green I real-time PCR performed with saturing concentrations of dye. In such conditions, centrally symmetric amplification curves are expected. However, in our experience, we found several non-symmetric amplification curves shown to have good amplification efficiency using standard curve analysis (Additional file 1 and 3). In order to find a suitable mathematical representation of the complete PCR kinetic curve we compared the standard error of estimate obtained by several equations that generate S-shaped curves (Tab. 1). As shown in Figure 1, these results demonstrated that real-time PCR readouts can be effectively modelled using the 5-parameter Richards function (Eq. 3). The Richards equation is an extension of the sigmoidal growth curve; specifically, when d coefficient is equal to 1, the sigmoidal and Richards curves are the same. Hence, we analysed the variation of the d coefficient in the presence of different input DNA and PCR efficiencies. Figure 3 shows that the d value is close to 1 at amplification mix percentages ranging from 100% to 90% while at lower amplification mix contents, where PCR efficiency decreases, the d coefficient was significantly higher than 1 regardless of the starting DNA content (Fig. 3; Tab. 2). These data demonstrate that sigmoidal fitting represents a good approximation of real-time PCR kinetic only in the presence of optimal amplification conditions while the Richards curve is more suited when PCR is inhibited. Since the Richards growth equation includes sigmoidal amplification curves, when d = 1, this nonlinear fitting was used in our method.
Figure 3Distribution of Richards coefficients (d) estimated from PCR fluorescence curves using Eq. 3 in nonlinear fitting procedure. Richards coefficient values were determined from 420 independent PCR reactions. The data have been reported in Log10scale, and represented as mean and standard deviation.
Table 1Comparison of five S-shaped models to fit the PCR curve: Sigmoid, Richards, Gompertz, Hill and Chapman. In this table, f is the fluorescence at cycle x; Fmax represents the maximum fluorescence value; Fb is the background reaction fluorescence; b, c and d determine the shape of each curve. For each model the determination coefficient (R2), the adjusted determination coefficient (Adj R2) and the standard error of estimate have been calculated.
|Name||Equation||Estimated Parameters||R2||Adj R2||Standard Error of Estimate|
|Sigmoid||f = Fb+Fmax/(1+exp(-(x-c)/b))||45.11||1.49||22.37||-0.03||1||1||0.1354|
|Richards||f = Fb+(Fmax/(1+exp(-(1/b)*(x-c)))^d)||45.11||1.58||21.95||0.02||1.20||1||1||0.0926|
|Gompertz||f = Fb+Fmax *exp(-exp(-(x-c)/b))||45.19||2.15||21.45||0.29||0.9992||0.9992||0.6006|
|Hill||f = Fb+Fmax *x^b/(d^b+x^b)||45.18||14.95||0.08||22.34||1||1||0.1351|
|Chapman||f = Fb+Fmax *(1-exp(-b*x))^d||45.19||0.46||0.29||20615||0.9992||0.9992||0.6006|
Table 2tstatistic values obtained for all variable combinations. When t < 0 the Richards coefficient is lower than 1, while for t > 0 the Richards coefficient is higher than 1.
|Amplification mix percentage|
Figure 4Plot of fluorescence observations versus cycle number obtained from the same starting DNA but in presence of decreasing amounts of amplification mix. This slight PCR inhibition produces curves which are less steep than controls and shifted towards the right. When analysed by the threshold method, these curves showed higher Ct values with a CV% of 1.45% (A). An example of Cy0 procedure has been reported for the same data set (B). In this method, the amplification reactions are described by the tangent crossing the inflection point of fluorescence curves. As shown in this figure, the straight-lines originating from PCRs, characterized by slightly different PCR efficiency and the same starting amounts, tend to cross into a common point near the x-axis leading to small variations in the Cy0 values (CV% = 0.6%).
Precision and accuracy of the Ct, Cp and Cy0 methods.The performance of the Ct, Cp and Cy0 methods was compared in terms of precision and accuracy over a wide range of input DNA concentrations and under different reaction efficiencies obtained by decreasing the amount of amplification mix as reported in Liu and Saint [18, 27]. As shown in Figure 5A, the Ct method is highly rigorous at maximum reaction efficiency regardless of the starting DNA template. However, the absolute value of RE increased almost linearly with the decrease of efficiency regardless of the template concentrations resulting in an underestimation of the unknown of about 50% at the lowest amplification efficiencies. The Cp was more accurate than the Ct method in the presence of different amounts of amplification mix. Indeed, the relative error in the presence of 100% amplification mix tended towards zero as it did using the Ct method. However, when the efficiency declined, the RE increased initially in the same manner at low and high input DNA concentrations, while at 60-70% of the amplification mix, this method markedly underestimated at low concentrations (mean RE60% mix; = -0.58; Fig. 5C). Finally, the Cy0 method was more accurate than the Cp method (mean RE -0.12 versus -0.18, respectively; Fig. 5C, E), which in turn was better than the Ct method (mean RE = -0.31). Notably, at optimal amplification conditions (90-100% of the amplification mix) the Cp and Cy0 methods were equivalent, but at decreasing efficiencies, the Cy0 accuracy was more stable than that of the Cp in the concentration range from 3.14x107 to 3.14x105 molecules. At lower DNA concentrations, from 3.14x104 to 3.14x102 molecules, the RE proportionally increased with the efficiency decline, but this underestimate was less marked than that of the Cp method at the same starting DNA (Fig. 5C, E). Regarding the precision of the three methods, the variation coefficients were determined for each combination of initial template amount and amplification mix percentage. The random error of quantification achieved by the Cp and Cy0 method was similar (mean CV% 21.8% and 22.5%, respectively), while the Ct procedure produced an overall CV% of about 39.7% (Tab. 3). When the CV was analysed in relation to PCR efficiency and input DNA, an area of low variation coefficients for the three methods was found between 3.14x104 and 3.14x107 molecules as starting material (Fig. 5B, D, F). With DNA amounts ranging from 3.14x103 to 3.14x102 molecules, the precision progressively decreased in each analysis procedure. These variations were not efficiency-dependent, but were related to initial DNA quantity as shown by the shapes of level curves reported in figure 5B, D and F, which were perpendicular to the input template amounts.
Figure 5Comparison of the Ct, Cp and Cy0 methods in terms of precision and accuracy. The accuracy of each method has been reported as Relative Error (RE = expected value ? estimated value) while the precision was evaluated measuring the variation coefficient (CV%). The 3D plots show the variation of relative error in relation to amplification mix percentage and log10 input DNA for the Ct (A), Cp (C) and Cy0 (E) methods. The areas in the level curve graphs represent the CV% values obtained for each amplification mix percentage and Log10 input DNA combination using the Ct (B), Cp (D) and Cy0 (F) methods.
Table 3Comparison of mean Relative Error and mean Variation Coefficient among the Ct, Cp, Cy0 and SCF methods. The reported data were calculated on 420 PCRs except for a) in which the reaction number was 210.
Experimental system 2: Real-time PCR quantification in the presence of the inhibitor IgGThe real-time amplification plot of 4.05x106 DNA molecules with increasing concentrations of IgG demonstrates the effects of PCR inhibition on amplification efficiency and accumulated fluorescence (Fig. 6A). As inhibitor concentrations increased, the amplification curves showed lower plateau fluorescence levels and a shift towards the right and the bottom of the inflection points, leading to amplification curves that were less steep and not as symmetric as those obtained in absence of the inhibitor agent (Fig. 6A). As shown in figure 6A the amplification curves inhibited by IgG showed a shape very similar to those resulting from the system of amplification mix reduction (system 1; Fig. 4A). Quantitative data analysis of these amplification plots showed that the estimated DNA quantities were systematically underestimated in the presence of IgG concentrations higher than 0.25 µg/ml and 1 µg/ml using Ct and Cp methods, respectively. However, the Cy0 method was able to adjust this bias minimizing the RE at high IgG concentrations (RE = 4.98%; CV = 4.33%; Fig. 6B). Furthermore, in presence of high IgG concentrations, the SCF approach, modified according to Rutledge 2004 , was inapplicable because it was impossible to minimize F0 value (Additional file 5).
DiscussionNone of the current quantitative PCR data treatment methods is in fact fully assumption-free, and their statistical reliability are often poorly characterized. In this study, we evaluated whether known real-time elaboration methods could estimate the amount of DNA in biological samples with precision and accuracy when reaction efficiencies of the unknown are different from those of the standard curve. Our experimental systems consisted in the quantification of samples with the same known starting template amount but the amplification reaction, performed for the real-time PCR assay, had a slightly decreasing efficiency. This is clearly not in agreement with the main assumption of the threshold approach which holds that the amplification efficiency of samples has to be identical to, or not significantly different from, that predicted by the standard curve. However, such an assumption has been reported to be patently invalid for many cases in medical diagnostics. In fact, some, if not all, of the biological samples may contain inhibitors that are not present in the standard nucleic acid samples used to construct the calibration curve, leading to an underestimation of the DNA quantities in the unknown samples [28, 29].
Figure 6Real-time PCR amplification plots obtained from the same starting DNA in the presence of IgG acting as reaction inhibitor This inhibition system produces curves which are progressively less steep than non-inhibited reactions with increasing IgG concentrations (A). When analysed by the Ct, Cp and Cy0 methods these curves showed a RE% of -25.37%, -9.02% and 4.98% and a CV% of 25.62%, 10.66% and 4.33%%, respectively (B).
ConclusionsReal-time PCR analysis is becoming increasingly important in biomedical research because of its accuracy, sensitivity and high efficiency. Although, real-time PCR analysis has gained considerable attention, it is far from being a standard assay. The standard methods are quite stable and straightforward but the accuracy of estimates is strongly impaired if efficiency is not equal in all reactions. Furthermore, the assumption of uniform efficiency has been reported to be invalid in many cases regarding medical diagnostics. In fact, the biological samples may contain inhibitors that could lead to different amplification efficiencies among samples. We propose, in this report, a modified standard curve-based method, called Cy0, that does not require the assumption of uniform reaction efficiency between standards and unknown.
List of abbreviations usedCp: crossing point; Ct: threshold cycle; CV: coefficient of variation; IgG: immunoglobulin G; RE: relative error; SCF: sigmoidal curve fitting.
Authors' contributionsMG and DS carried out the design of the study, participated in data analysis, developed the Cy0 method and drafted the manuscript. MBLR participated in data collection and analysis and critically revised the manuscript. LS carried out the real-time PCR. VS participated in the design of the study and critically revised the manuscript. All authors read and approved the final manuscript.
AcknowledgementsWe thank Dr. Pasquale Tibollo for technical assistance and Dr. Giosué Annibalini for helpful comments on the manuscript.
1. Higuchi R, Fockler C, Dollinger G, Watson R: Kinetic PCR analysis: real-time monitoring of DNA amplification reactions. Biotechnology (N Y) 1993, 11(9):1026-1030.
2. Schmittgen TD: Real-time quantitative PCR. Methods 2001, 25(4):383-385.
3. Bustin SA, Nolan T: Pitfalls of quantitative real-time reverse-transcription polymerase chain reaction. J Biomol Tech 2004, 15(3):155-166.
4. Gingeras TR, Higuchi R, Kricka LJ, Lo YM, Wittwer CT: Fifty years of molecular (DNA/RNA) diagnostics. Clin Chem 2005, 51(3):661-671.
5. Bustin SA, Mueller R: Real-time reverse transcription PCR (qRT-PCR) and its potential use in clinical diagnosis. Clin Sci (Lond) 2005, 109(4):365-379.
6. Nolan T, Hands RE, Bustin SA: Quantification of mRNA using real-time RT-PCR. Nature Protocols 2006, 1(3):1559-1582.
7. Marubini E, Verderio P, Raggi CC, Pazzagli M, Orlando C: Statistical diagnostics emerging from external quality control of real-time PCR. The International journal of biological markers 2004, 19(2):141-146.
8. Pfaffl M: Quantification strategies in real time PCR. In A-Z of quantitative PCR Edited by: Bustin SA La Jolla, CA, International University Line 2004.
9. Luu-The V, Paquet N, Calvo E, Cumps J: Improved real-time RT-PCR method for high-throughput measurements using second derivative calculation and double correction. BioTechniques 2005, 38(2):287-293.
10. Livak KJ: ABI Prism 7700 sequence detection system. User Bulletin 2. PE Applied Biosystems 1997.
11. Rutledge RG, Cote C: Mathematics of quantitative kinetic PCR and the application of standard curves. Nucleic Acids Res 2003, 31(16):e93.
12. Raeymaekers L: A commentary on the practical applications of competitive PCR. Genome Res 1995, 5(1):91-94.
13. Bar T, Stahlberg A, Muszta A, Kubista M: Kinetic Outlier Detection (KOD) in real-time PCR. Nucleic Acids Res 2003, 31(17):e105.
14. Lefevre J, Hankins C, Pourreaux K, Voyer H, Coutlee F: Prevalence of selective inhibition of HPV-16 DNA amplification in cervicovaginal lavages. Journal of medical virology 2004, 72(1):132-137.
15. Sunen E, Casas N, Moreno B, Zigorraga C: Comparison of two methods for the detection of hepatitis A virus in clam samples (Tapes spp.) by reverse transcription-nested PCR. International journal of food microbiology 2004, 91(2):147-154.
16. Jiang J, Alderisio KA, Singh A, Xiao L: Development of procedures for direct extraction of Cryptosporidium DNA from water concentrates and for relief of PCR inhibitors. Applied and environmental microbiology 2005, 71(3):1135-1141.
17. Kontanis EJ, Reed FA: Evaluation of real-time PCR amplification efficiencies to detect PCR inhibitors. J Forensic Sci 2006, 51(4):795-804.
18. Liu W, Saint DA: Validation of a quantitative method for real time PCR kinetics. Biochem Biophys Res Commun 2002, 294(2):347-353.
19. Goll R, Olsen T, Cui G, Florholmen J: Evaluation of absolute quantitation by nonlinear regression in probe-based real-time PCR. BMC Bioinformatics 2006, 7:107.
20. Karlen Y, McNair A, Perseguers S, Mazza C, Mermod N: Statistical significance of quantitative PCR. BMC Bioinformatics 2007, 8:131.
21. Bustin SA: Absolute quantification of mRNA using real-time reverse transcription polymerase chain reaction assays. J Mol Endocrinol 2000, 25(2):169-193.
22. Tichopad A, Didier A, Pfaffl MW: Inhibition of real-time RT-PCR quantification due to tissue-specific contaminants. Mol Cell Probes 2004, 18(1):45-50.
23. Nolan T, Hands RE, Ogunkolade W, Bustin SA: SPUD: a quantitative PCR assay for the detection of inhibitors in nucleic acid preparations. Anal Biochem 2006, 351(2):308-310.
24. Ramakers C, Ruijter JM, Deprez RH, Moorman AF: Assumption-free analysis of quantitative real-time polymerase chain reaction (PCR) data. Neurosci Lett 2003, 339(1):62-66.
25. Richards F: A flexible growth function for empirical use. Journal of experimental Botany 1959, 10:290-300.
26. Rutledge RG: Sigmoidal curve-fitting redefines quantitative real-time PCR with the prospective of developing automated high-throughput applications. Nucleic Acids Res 2004, 32(22):e178.
27. Alvarez MJ, Vila-Ortiz GJ, Salibe MC, Podhajcer OL, Pitossi FJ: Model based analysis of real-time PCR data from DNA binding dye protocols. BMC Bioinformatics 2007, 8(1):85.
28. Meijerink J, Mandigers C, van de Locht L, Tonnissen E, Goodsaid F, Raemaekers J: A novel method to compensate for different amplification efficiencies between patient DNA samples in quantitative real-time PCR. J Mol Diagn 2001, 3(2):55-61.
29. Stahlberg A, Aman P, Ridell B, Mostad P, Kubista M: Quantitative real-time PCR method for detection of B-lymphocyte monoclonality by comparison of kappa and lambda immunoglobulin light chain expression. Clin Chem 2003, 49(1):51-59.
30. Feller W: On the logistic law of growth and its empirical verification in biology. Acta Bioth Ser A 1940(2):51-66.
31. Birch C: A new generalized logistic sigmoid growth equation compared with the Richards growth equation. Annals of Botany 1999, 83:713-723.
32. Yin X, Goudriaan J, Lantinga EA, Vos J, Spiertz HJ: A flexible sigmoid function of determinate growth. Ann Bot (Lond) 2003, 91(3):361-371.
33. Lalam N: Estimation of the reaction efficiency in polymerase chain reaction. J Theor Biol 2006, 242(4):947-953.
34. Zhao S, Fernald RD: Comprehensive algorithm for quantitative real-time polymerase chain reaction. J Comput Biol 2005, 12(8):1047-1064.
35. Gevertz JL, Dunn SM, Roth CM: Mathematical model of real-time PCR kinetics. Biotechnol Bioeng 2005, 92(3):346-355.
36. Peirson SN, Butler JN, Foster RG: Experimental validation of novel and conventional approaches to quantitative real-time PCR data analysis. Nucleic Acids Res 2003, 31(14):e73.
37. Tichopad A, Dilger M, Schwarz G, Pfaffl MW: Standardized determination of real-time PCR efficiency from a single reaction set-up. Nucleic Acids Res 2003, 31(20):e122.
38. Liu W, Saint DA: A new quantitative method of real time reverse transcription polymerase chain reaction assay based on simulation of polymerase chain reaction kinetics. Anal Biochem 2002, 302(1):52-59.
39. Livak KJ, Flood, S.J., Marmaro, J., Giusti, W., Deetz, K.: Oligonucleotides with fluorescent dyes at opposite ends provide a quenched probe system useful for detecting PCR product and nucleic acid hybridization. PCR Methods appl 1995, 4:357-362.